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The topology of 4-manifolds. (English) Zbl 0668.57001

Lecture Notes in Mathematics, 1374. Berlin etc.: Springer-Verlag. vi, 108 p. DM 25.00 (1989).
The book opens with the following words by the author: “When I began to think about 4-manifolds in 1973, the basic theorems included the Whitehead-Milnor theorem on homotopy type, Rohlin’s theorem, \(\Omega_ 4^{SO}=Z\), \(\Omega_ 4^{spin}=Z\), the Hirzebruch index theorem \(p_ 1=3\sigma\), and the Wall’s theorem on diffeomorphisms and h-cobordism. These theorems were untranslated (Rohlin) or unreadable (Whitehead), or were special cases of big machines in algebraic topology \((\Omega_ 4^{SO}=Z\), \(\Omega_ 4^{spin}=Z\), \(p_ 1=3\sigma)\), or, even though accessible, could, with hindsight, use streamlining (Wall).” In the author’s opinion “the algebraic topological proofs are powerful, and beautiful mathematics in their own right, but there ought to be proofs of the fundamental 4-manifold theorems which belong to the field of 4-dimensions (or less), and prepare the student in the geometric side of the theory”. This is exactly what the author has achieved in this book: he presents all the above mentioned classical results on 4-manifolds with geometric proofs, using handlebody theory. Most of the arguments are either new or are refurbishings of post proofs. The last three chapters include an introduction to Casson handles and Freedman’s work as well as some heretofore unpublished work on exotic \(R^ 4\)’s (e.g. R. Gompf’s simple construction of countably many exotic \(R^ 4\)’s which he discovered “in fall 1984 during tea at MSRI”). The author has done a great job in presenting this difficult subject. The book will be most valuable to anyone who has a good knowledge of smooth manifolds and characteristic classes (in low dimensions) and wants to study 4-manifolds. A suggested companion volume should be the other monograph on topology of 4-manifolds by M. H. Freedman and F. S. Quinn [Topology of 4-manifolds (1988)].
Reviewer: D.Repovš

MSC:

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
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